86 HISTORY OF THE THEORY or NUMBERS. [CHAP, in where d ranges over those divisors of a^—1 which do not divide a' — l for Q<v<N', while, in the former, pit. . ., p8 are the distinct prime factors of N, and n is prune to N. L. Gegenbauer172 wrote F(at n) in the form 2v(d)an/d, where d ranges over the divisors of n, and /z(d) is the function discussed in Chapter XIX on Inversion. As there shown, S/z(d) =0 if n>l. This case f(x) =IJL(X) is used to prove the generalization: If the function /(a) has the property that S/(d) is divisible by n, then for every integer a the function 2f(d)an/d is divisible by n, where hi each sum d ranges over the divisors of n. Another special case, f(x) =<£(#)> was noted by MacMahon.112 J. Westlund173 considered any ideal A in a given algebraic number field, the distinct prune factors P1} . . . , P< of A, the norm n(A) of A, and proved that if a is any algebraic integer, is always divisible by A. J. Vdlyi174 noted that the number of triangles similar to their nth pedal but not to the dth pedal (d<n) is ^ Pi> Z>2>- - • are the distinct prime factors of n, and \t/(k)=2k(2k — 1). He proved that x W is divisible by n, since if the nth pedal to ABC is the first one similar to ABC, a like property is true of the first pedal, . . ., (n — l)th pedal, so that the x W triangles fall into sets of n each of period n. [Note A. Axer175 proved the folio wing generalization of Gegenbauer's172 theorem: If G(ri, . . . , PA) is any polynomial with integral coefficients, and if, when d ranges over all the divisors of n, for a particular function G = GQ and a particular set of values r10, . . ., rM, not a set of solutions of 00, and for which GQ is prime to n, then it holds for every G and every set r^ . . . , rh. FUHTHER GENERALIZATIONS OF FERMAT'S THEOREM. For the generalization to Galois imaginaries, see Ch. VIII. For the generalization by Lucas, see Ch. XVII, Lucas,39 Carmichael.89 On ofel (mod n) for x prime to n, see Cauchy,26 Moreau,93 Epstein/12 of Ch. VII. 0. H. Mitchell178 considered the 2*' products s of distinct primes dividing fc = p?...p? and denoted by T.(fc) the number of positive integers Xa<k which are divisible by s but by no prime factor of k not dividing s. 172Monatshefte Math. Phys., 11, 1900, 287-8. mProc. Indiana Ac. Sc., 1902, 78-79. 174Monatshefte.Math. Phys., 14, 1903, 243-253. 175Monatshefte Math. Phys., 22, 1911, 187-194. 178Amer. Jour. Math., 3, 1880, 300; Johns Hopkins Univ. Circular, 1, 1880-1, 67, 97.